Relaxação e decaimento para pontos de equilíbrio em mapeamentos não lineares unidimensionais

We investigate in this work the behaviour of the decay to the fixed points, in particular along the bifurcations, for a family of one-dimensional logistic-like discrete mappings. We start with the logistic map focusing in the transcritical bifurcation. Next we investigate the convergence to the stationary state at the cubic map. At the end we generalise the procedure for a mapping of the logistic-like type. Near the fixed point, the dynamical variable varies slowly. This property allows us to approximate/rewrite the equation of differences, hence natural from discrete mappings, into an ordinary differential equation. We then solve such equation which furnishes the evolution towards the stationary state. Our numerical simulations confirm the theoretical results validating the above mentioned approximation

The 2:1 cycloadducts from [3+2] 1,3-dipolar cycloaddition of nitrile oxide and vinylacetic acid. Synthesis and liquid crystal behaviour

Four liquid crystals (LC) 3,7a-bis(4-alkyloxyphenyl)-7,7a-dihydro-6H-isoxazolo[2,3-d][1,2,4]oxadiazol-6-yl)acetic acid (7a-d) were synthesised and the mesomorphic behaviour reported. The LCs were characterised as 2: 1 bisadducts, which were obtained from a double [3+2] 1,3-dipolar cycloaddition. In the first step, the cycloaddition of 4-alkyloxyphenylnitrile oxide (4a-d) and vinylacetic acid (5) gave the initial unobserved 1:1 cycloadducts 2-[3-(4-alkyloxyphenyl)-4,5-dihydroisoxazol-5-yl]acetic acid (6a-d). In the second step, the addition of a second equivalent of 4 to 6 yielded the 2: 1 bisadducts 7a-d without any traces of 6. All compounds 7a-d were unstable during the transition from the mesophase to the isotropic state upon first heating as evidenced by the large peaks in the differential scanning calorimetry traces. Due to the chemical instability of the compounds upon heating, the transition temperature related to the smectic C to smectic A transitions was acquired by means of an image processing method. X-Ray diffraction experiments were also used to analyse the liquid-crystalline phases. A theoretical calculation was performed using density functional theory (DFT) methods at the PBE1PBE/6-311+G(2d,p) level (with solvent effect) in order to get information about the energetic profile of the 2: 1 cycloaddition. DFT studies revealed that the cycloaddition process is controlled by the HOMO(dipolarophile) - LUMO(1,3-dipole), and that the double [3+2] 1,3-dipolar cycloaddition reaction is quite possible.

Isoxazolidines and oxazines: preliminary studies for the synthesis of N,O-heterocyclic systems via an organocatalyzed 1,3-Dipolar Cycloaddiction and a tandem reaction mediated by TEMPO salts

This thesis is the result of the study of two reactions leading to the formation of important heterocyclic compounds of potential pharmaceutical interest. The first study concerns the reaction of (1,3)-dipolar cycloaddition between nitrones and activated olefins by hydrogen bond catalysis of thioureas derivatives leading to the formation of a five-membered cyclic adducts, an interesting and strategic synthetic intermediate, for the synthesis of benzoazepine. The second project wants to explore the direct oxidative C(sp3)-H α-alkylation of simple amides with subsequent addition of an olefin and cyclization in order to obtain the corresponding oxazine. Both reactions are still under development.

13C NMR of a single molecule magnet: analysis of pseudocontact shifts and residual dipolar couplings

Paramagnetic triple decker complexes of lanthanides are promising Single Molecule Magnets (SMMs), with many potential uses. Some of them show preferable relaxation behavior, which enables the recording of well resolved NMR spectra. These axially symmetric complexes are also strongly magnetically anisotropic, and this property can be described with the axial component of the magnetic susceptibility tensor, χa. For triple decker complexes with phthalocyanine based ligands, the Fermi˗contact contribution is small. Hence, together with the axial symmetry, the experimental chemical shifts in 1H and 13C NMR spectra can be modeled easily by considering pseudocontact and orbital shifts alone. This results in the determination of the χa value, which is also responsible for molecular alignment and consequently for the observation of residual dipolar couplings (RDCs). A detailed analysis of the experimental 1H-13C and 1H-1H couplings revealed that contributions from RDCs (positive and negative) and from dynamic frequency shifts (negative for all observed couplings) have to be considered. Whilst the pseudocontact shifts depend on the average positions of 1H and 13C nuclei relative to the lanthanide ions, the RDCs are related to the mobility of nuclei they correspond to. This phenomenon allows for the measurement of the internal mobility of the various groups in the SMMs.

Regiocontrolled Synthesis of Pyrazole Derivatives Through 1,3-Dipolar Cycloaddition Reaction And Synthesis of Helicene-Thiourea based and Polymer Supported Soos's Catalyst for Asymmetric Synthesis

In first part we have developed a simple regiocontrolled protocol of 1,3-DC to get ring fused pyrazole derivatives. These pyrazole derivatives were synthesized using 1,3-DC between nitrile imine and various dipolarophiles such as alkynes, cyclic α,β-ketones, lactones, thiocatones and lactums. The reactions were found to be highly regiospecific. In second part we have discussed about helicene, its properties, synthesis and applications as asymmetric catalyst.Due to inherent chirality, herein we have made an attempt to synthesize the helicene-thiourea based catalyst for asymmetric catalysis. The synthesis involved formation of two key intermediates viz, bromo-phenanthrene 5 and a vinyl-naphthalene 10. The coupling of these two intermediates leads to formation of hexahelicene.

Nonequilibrium hysteresis and spin relaxation in the mixed-anisotropy dipolar-coupled spin-glass LiHo0.5Er0.5F4

We present a study of the model spin-glass LiHo0.5Er0.5F4 using simultaneous ac susceptibility, magnetization, and magnetocaloric effect measurements along with small angle neutron scattering (SANS) at sub-Kelvin temperatures. All measured bulk quantities reveal hysteretic behavior when the field is applied along the crystallographic c axis. Furthermore, avalanchelike relaxation is observed in a static field after ramping from the zero-field-cooled state up to 200–300 Oe. SANS measurements are employed to track the microscopic spin reconfiguration throughout both the hysteresis loop and the related relaxation. Comparing the SANS data to inhomogeneous mean-field calculations performed on a box of one million unit cells provides a real-space picture of the spin configuration. We discover that the avalanche is being driven by released Zeeman energy, which heats the sample and creates positive feedback, continuing the avalanche. The combination of SANS and mean-field simulations reveal that the conventional distribution of cluster sizes is replaced by one with a depletion of intermediate cluster sizes for much of the hysteresis loop.

Molecular dynamics of a grid-mounted molecular dipolar rotor in a rotating electric field

Classical molecular dynamics is applied to the rotation of a dipolar molecular rotor mounted on a square grid and driven by rotating electric field E(ν) at T ≃ 150 K. The rotor is a complex of Re with two substituted o-phenanthrolines, one positively and one negatively charged, attached to an axial position of Rh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{2}^{4+}}}\end{equation*}\end{document} in a [2]staffanedicarboxylate grid through 2-(3-cyanobicyclo[1.1.1]pent-1-yl)malonic dialdehyde. Four regimes are characterized by a, the average lag per turn: (i) synchronous (a < 1/e) at E(ν) = |E(ν)| > Ec(ν) [Ec(ν) is the critical field strength], (ii) asynchronous (1/e < a < 1) at Ec(ν) > E(ν) > Ebo(ν) > kT/μ, [Ebo(ν) is the break-off field strength], (iii) random driven (a ≃ 1) at Ebo(ν) > E(ν) > kT/μ, and (iv) random thermal (a ≃ 1) at kT/μ > E(ν). A fifth regime, (v) strongly hindered, W > kT, Eμ, (W is the rotational barrier), has not been examined. We find Ebo(ν)/kVcm−1 ≃ (kT/μ)/kVcm−1 + 0.13(ν/GHz)1.9 and Ec(ν)/kVcm−1 ≃ (2.3kT/μ)/kVcm−1 + 0.87(ν/GHz)1.6. For ν > 40 GHz, the rotor behaves as a macroscopic body with a friction constant proportional to frequency, η/eVps ≃ 1.14 ν/THz, and for ν < 20 GHz, it exhibits a uniquely molecular behavior.

Binap-gold (I) versus Binap-silver trifluoroacetate complexes as catalysts in 1,3-dipolar cycloadditions of axomethine ylides

The 1,3-dipolar cycloaddition between azomethine ylides and alkenes is efficiently catalysed by [{(Sa)-Binap-Au(tfa)}2] (Binap=2,2′-bis(diphenylphosphino)-1,1′-binaphthyl; tfa=trifluoroacetyl). Maleimides, 1,2-bis(phenylsulfonyl)ethylene, chalcone and nitrostyrene were suitable dipolarophiles even when using sterically hindered 1,3-dipole precursors. The results obtained in these transformations improve the analogous ones obtained in the same reactions catalysed by [Binap–Ag(tfa)]. In addition, computational studies have also been carried out to demonstrate both the high enantioselectivity exhibited by the chiral gold(I) complex, and the non-linear effect observed in this transformation.

Enantioselective 1,3-dipolar cycloadditions of azlactones and electrophilic alkenes catalyzed by dimeric BinapAuTFA complexes

Glycine-derived azlactones react with maleimides using (S)- or (R)-dimeric BinapAuTFA complexes affording the corresponding cycloadducts in good yields and high enantioselections (up to 99% ee). The intermediate carboxylic acids are treated with trimethylsilyldiazomethane and isolated as Δ¹-pyrroline methyl esters. These cycloadducts are transformed into exo-proline derivatives by reduction with NaBH3CN in acidic media. On the other hand, N-benzoylalanine-derived oxazolone reacts with tert-butyl acrylate providing the cycloadduct with the ester group at the 3-position with a trans-relative configuration with respect to the methyl ester group.

Kinetic study of thermal 1,3-dipolar cycloaddition of azomethine ylides using differential scanning calorimetry as monitoring window

Kinetics of 1,3-dipolar cycloaddition involving azomethine ylides, generated from thermal [1,2]-prototropy of the corresponding imino ester, employing differential scanning calorimetry (DSC), is surveyed. Glycine and phenylalanine derived imino esters have different behavior. The first one prefers reacting with itself at 75 ºC, rather than with the dipolarophile. However, the α-substituted imino ester gives the cycloadduct at higher temperatures. The thermal dynamic analysis by 1H NMR of the neat reaction mixture of the glycine derivative reveals the presence of signals corresponding to the dipole in very small proportion. The non-isothermal and isothermal DSC curves of the cycloaddition of phenylalaninate and diisobutyl fumarate are obtained from freshly prepared samples. The application of known kinetic models and mathematical multiple non-linear regressions (NLR) allow to determine and to compare Ea, lnA, reaction orders, and reaction enthalpy. Finally a rate equation for each different temperature can be established for this particular thermal cycloaddition.

Binap-silver salts as chiral-catalysts for the enantioselective 1,3-dipolar cycloaddition of azomethine ylides and alkenes

Binap-AgSbF6 catalyzed 1,3-dipolar cycloadditions between azomethine ylides and electrophilic alkenes are described and compared with analogous transformations mediated by other Binap-silver(I) salt complexes. Maleimides and 1,2-bis(phenylsulfonyl)ethylene are suitable dipolarophiles for obtaining very good enantioselectivities, even better values are generated by a multicomponent version. There are some very interesting applications of the disulfonylated cycloadducts in the total synthesis of cis-2,5-disubstituted pyrrolidines, precursors of natural products, or valuable intermediates in the synthesis of antiviral compounds.